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Series: Combinatorics Seminar

A classical theorem of Spencer shows that any set system with n sets and n elements admits a coloring of discrepancy O(n^1/2). Recent exciting work of Bansal, Lovett and Meka shows that such colorings can be found in polynomial time. In fact, the Lovett-Meka algorithm finds a half integral point in any "large enough" polytope. However, their algorithm crucially relies on the facet structure and does not apply to general convex sets. We show that for any symmetric convex set K with measure at least exp(-n/500), the following algorithm finds a point y in K \cap [-1,1]^n with Omega(n) coordinates in {-1,+1}: (1) take a random Gaussian vector x; (2) compute the point y in K \cap [-1,1]^n that is closest to x. (3) return y. This provides another truly constructive proof of Spencer's theorem and the first constructive proof of a Theorem of Gluskin and Giannopoulos.

Series: Combinatorics Seminar

The Frankl union-closed sets conjecture states that there exists an element
present in at least half of the sets forming a union-closed family. We
reformulate the conjecture as an optimization problem and present an
integer program to model it. The computations done with this program lead
to a new conjecture: we claim that the maximum number of sets in a
non-empty union-closed family in which each element is present at most a
times is independent of the number n of elements spanned by the sets if n
is greater or equal to log_2(a)+1. We prove that this is true when n is
greater or equal to a. We also discuss the impact that this new conjecture
would have on the Frankl conjecture if it turns out to be true.
This is joint work with Jonad Pulaj and Dirk Theis.

Series: Combinatorics Seminar

Joint work with Yinon Spinka.

Consider a random coloring of a bounded domain in the bipartite graph Z^d with the probability of each color configuration proportional to exp(-beta*N(F)), where beta>0, and N(F) is the number of nearest neighboring pairs colored by the same color. This model of random colorings biased towards being proper, is the antiferromagnetic 3-state Potts model from statistical physics, used to describe magnetic interactions in a spin system. The Kotecky conjecture is that in such a model with d >= 3, Fixing the boundary of a large even domain to take the color $0$ and high enough beta, a sampled coloring would typically exhibits long-range order. In particular a single color occupies most of either the even or odd vertices of the domain. This is in contrast with the situation for small beta, when each bipartition class is equally occupied by the three colors. We give the first rigorous proof of the conjecture for large d. Our result extends previous works of Peled and of Galvin, Kahn, Randall and Sorkin, who treated the zero beta=infinity case, where the coloring is chosen uniformly for all proper three-colorings. In the talk we shell give a glimpse into the combinatorial methods used to tackle the problem. These rely on structural properties of odd-boundary subsets of Z^d. No background in statistical physics will be assumed and all terms will be thoroughly explained.

Series: Combinatorics Seminar

Both for random words or random permutations, I will present a panoramic view of results on the (asymptotic) behavior of the length of the longest common subsequences . Starting with, now, classical results on expectations dating back to the nineteen-seventies I will move to recent results obtained by Ümit Islak and myself giving the asymptotic laws of this length and as such answering a decades-old well know question.

Series: Combinatorics Seminar

We present an algebraic framework which simultaneously generalizes
the notion of linear subspaces, matroids, valuated matroids, and oriented
matroids. We call the resulting objects matroids over hyperfields. We give
"cryptomorphic" axiom systems for such matroids in terms of circuits,
Grassmann-Plucker functions, and dual pairs, and establish some basic
duality theorems.

Series: Combinatorics Seminar

We show that there exists an absolute constant c>0 with the following property. Let A be a set in a finite field with q elements. If |A|>q^{2/3-c}, then the set (A-A)(A-A) consisting of products of pairwise differences of elements of A contains at least q/2 elements. It appears that this is the first instance in the literature where such a conclusion is reached for such type sum-product-in-finite-fileds questions for sets of smaller cardinality than q^{2/3}. Similar questions have been investigated by Hart-Iosevich-Solymosi and Balog.

Series: Combinatorics Seminar

Seymour and, independently, Kelmans conjectured in the 1970s that
every 5-connected nonplanar graph contains a subdivision of $K_5$. This
conjecture was proved by Ma and Yu for graphs containing $K_4^-$. Recently,
we proved this entire Kelmans-Seymour conjecture. In this talk, I will give
a sketch of our proof, and discuss related problems.
This is joint work with Dawei He and Xingxing Yu.

Series: Combinatorics Seminar

I will give a broad overview of the Hard Lefschetz property and the Hodge-Riemann relations in the theory of polytopes, complex manifolds, invariants, algebraic varieties, and tropical varieties.

Series: Combinatorics Seminar

Joint work with Shachar Lovett.

Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,\Sigma), where each element x \in X lies in t randomly selected sets of \Sigma, where t \le |X| is an integer parameter. We provide new discrepancy bounds in this case. Specifically, we show that when |\Sigma| \ge |X| the hereditary discrepancy of (X,\Sigma) is with high probability O(\sqrt{t \log t}), matching the Beck-Fiala conjecture upto a \sqrt{\log{t}} factor. Our analysis combines the Lov{\'a}sz Local Lemma with a new argument based on partial matchings.

Series: Combinatorics Seminar

Represent a genome with an edge-labelled, directed graph
having maximum total degree two. We explore a number of questions
regarding genome rearrangement, a common mode of molecular evolution. In
the single cut-or-join model for genome rearrangement, a genome can
mutate in one of two ways at any given time: a cut divides a degree two
vertex into two degree one vertices while a join merges two degree one
vertices into one degree two vertex.
Fix a set of genomes, each having the same set of edge labels. The
number of ways for one genome to mutate into another can be computed in
polynomial time. The number of medians can also be computed in
polynomial time. While single cut-or-join is, computationally, the
simplest mathematical model for genome rearrangement, determining the
number of most parsimonious median scenarios remains #P-complete. We
will discuss these and other complexity results that arose from an
abstraction of this problem. [This is joint work with Istvan Miklos.]